The present invention relates to computed tomography (CT) imaging apparatus; and more particularly, to a method for reconstructing images from divergent beams of acquired image data.
In a current computed tomography system, an x-ray source projects a fan-shaped beam which is collimated to lie within an X-Y plane of a Cartesian coordinate system, termed the “imaging plane.” The x-ray beam passes through the object being imaged, such as a medical patient, and impinges upon an array of radiation detectors. The intensity of the transmitted radiation is dependent upon the attenuation of the x-ray beam by the object and each detector produces a separate electrical signal that is a measurement of the beam attenuation. The attenuation measurements from all the detectors are acquired separately to produce the transmission profile.
The source and detector array in a conventional CT system are rotated on a gantry within the imaging plane and around the object so that the angle at which the x-ray beam intersects the object constantly changes. A group of x-ray attenuation measurements from the detector array at a given angle is referred to as a “view” and a “scan” of the object comprises a set of views made at different angular orientations during one revolution of the x-ray source and detector. In a 2D scan, data is processed to construct an image that corresponds to a two dimensional slice taken through the object. The prevailing method for reconstructing an image from 2D data is referred to in the art as the filtered backprojection technique. This process converts the attenuation measurements from a scan into integers called “CT numbers” or “Hounsfield units”, which are used to control the brightness of a corresponding pixel on a display.
The term “generation” is used in CT to describe successively commercially available types of CT systems utilizing different modes of scanning motion and x-ray detection. More specifically, each generation is characterized by a particular geometry of scanning motion, scanning time, shape of the x-ray beam, and detector system.
As shown in FIG. 1, the first generation utilized a single pencil x-ray beam and a single scintillation crystal-photomultiplier tube detector for each tomographic slice. After a single linear motion or traversal of the x-ray tube and detector, during which time 160 separate x-ray attenuation or detector readings are typically taken, the x-ray tube and detector are rotated through 1° and another linear scan is performed to acquire another view. This is repeated typically to acquire 180 views.
A second generation of devices developed to shorten the scanning times by gathering data more quickly is shown in FIG. 2. In these units a modified fan beam in which anywhere from three to 52 individual collimated x-ray beams and an equal number of detectors are used. Individual beams resemble the single beam of a first generation scanner. However, a collection of from three to 52 of these beams contiguous to one another allows multiple adjacent cores of tissue to be examined simultaneously. The configuration of these contiguous cores of tissue resembles a fan, with the thickness of the fan material determined by the collimation of the beam and in turn determining the slice thickness. Because of the angular difference of each beam relative to the others, several different angular views through the body slice are being examined simultaneously. Superimposed on this is a linear translation or scan of the x-ray tube and detectors through the body slice. Thus, at the end of a single translational scan, during which time 160 readings may be made by each detector, the total number of readings obtained is equal to the number of detectors times 160. The increment of angular rotation between views can be significantly larger than with a first generation unit, up to as much as 36°. Thus, the number of distinct rotations of the scanning apparatus can be significantly reduced, with a coincidental reduction in scanning time. By gathering more data per translation, fewer translations are needed.
To obtain even faster scanning times it is necessary to eliminate the complex translational-rotational motion of the first two generations. As shown in FIG. 3, third generation scanners therefore use a much wider fan beam. In fact, the angle of the beam may be wide enough to encompass most or all of an entire patient section without the need for a linear translation of the x-ray tube and detectors. As in the first two generations, the detectors, now in the form of a large array, are rigidly aligned relative to the x-ray beam, and there are no translational motions at all. The tube and detector array are synchronously rotated about the patient through an angle of 180–360°. Thus, there is only one type of motion, allowing a much faster scanning time to be achieved. After one rotation, a single tomographic section is obtained.
Fourth generation scanners feature a wide fan beam similar to the third generation CT system as shown in FIG. 4. As before, the x-ray tube rotates through 360° without having to make any translational motion. However, unlike in the other scanners, the detectors are not aligned rigidly relative to the x-ray beam. In this system only the x-ray tube rotates. A large ring of detectors are fixed in an outer circle in the scanning plane. The necessity of rotating only the tube, but not the detectors, allows faster scan time.
Divergent fan-beam scanning modes have the potential to allow fast data acquisition. But image reconstruction from divergent-beam projections poses a challenge. In particular, the projection-slice theorem is not directly applicable to the divergent-beam projections since the shift-invariance in a single view of projections is lost in the divergent-beam case. One way to bypass this problem is to explicitly rebin the measured divergent-beam projections into parallel beam projections. This is the basic method currently used in solving the problem of fan-beam image reconstruction. After the rebinning process, one can take the advantages of the fast Fourier transforms (FFT) to efficiently reconstruct images.
Recently, the divergent cone-beam reconstruction problem in x-ray CT has attracted increased attention due to the rapid development of multi-row detectors. In the cone-beam case, it is much more complicated to rebin cone-beam projections into parallel-beam projections. The huge cone-beam data set also poses a big challenge to the potential data storage during the rebinning process. The main stream of the developments in cone-beam reconstruction has been focused on the development of approximate or exact reconstruction methods. For circular-based source trajectories, methods disclosed by L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical Cone Beam Algorithm,” J. Opt. Soc. Am. A 1, 612–619(1984); G. Wang, T. H. Lin, P. Cheng, and D. M. Shinozaki, “A general cone-beam reconstruction algorithm,” IEEE Trans. Med. Imaging 12, 486–496 (1993); generate acceptable image quality up to moderate cone angles (up to 10° or so). Exact reconstruction algorithms have also been proposed and further developed for both helical source trajectory and more general source trajectories. Most recently, a mathematically exact and shift-invariant FBP reconstruction formula was proposed for the helicauspiral source trajectory A. Katsevich, “Theoretically exact filtered backprojection-type inversion algorithm for spiral CT,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 2012–2026 (2002).
There are two types of analytic cone-beam reconstruction algorithms: exact and approximate. The first and most practical approximate cone-beam reconstruction algorithm was proposed by Feldkamp, Davis and Kress (FDK) for a circular x-ray source trajectory. It is a heuristic extension of standard fan-beam reconstruction to the cone-beam case by introducing a “cosine” factor. Extensions to this approximate algorithm have been developed. Due to its one-dimensional shift invariant filtering kernel it has been applied to various medical imaging systems. The original FDK algorithm is applied to a complete circular scanning path. Recently, the fan-beam super-short-scan FBP reconstruction formulae has been extended to the cone-beam case to obtain super-short-scan FDK-type algorithms.
There is an alternative method to obtain an approximate cone-beam reconstruction algorithm; namely, apply a mathematically exact algorithm to an incomplete source trajectory. In the case of a single arc used to reconstruct a 3D volume the data is incomplete according to Tuy's data sufficiency condition. The problem has been addressed in the Grangeat framework using a shift variant filtered backprojection reconstruction formula. When a rebinning scheme is introduced, a short scan image reconstruction algorithm in the Grageat framework results.
Recently, a general procedure to generate shift-invariant cone-beam reconstruction algorithms has been developed for a general source trajectory provided that the data sufficiency condition is fulfilled. There is a need for such an image reconstruction method where the data sufficiency condition has not been met.